Integrand size = 37, antiderivative size = 126 \[ \int \frac {1}{(c e+d e x)^{9/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=-\frac {2 \sqrt {1-c^2-2 c d x-d^2 x^2}}{7 d e (c e+d e x)^{7/2}}-\frac {10 \sqrt {1-c^2-2 c d x-d^2 x^2}}{21 d e^3 (c e+d e x)^{3/2}}+\frac {10 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c e+d e x}}{\sqrt {e}}\right ),-1\right )}{21 d e^{9/2}} \]
10/21*EllipticF((d*e*x+c*e)^(1/2)/e^(1/2),I)/d/e^(9/2)-2/7*(-d^2*x^2-2*c*d *x-c^2+1)^(1/2)/d/e/(d*e*x+c*e)^(7/2)-10/21*(-d^2*x^2-2*c*d*x-c^2+1)^(1/2) /d/e^3/(d*e*x+c*e)^(3/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.03 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.32 \[ \int \frac {1}{(c e+d e x)^{9/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=-\frac {2 (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {7}{4},\frac {1}{2},-\frac {3}{4},(c+d x)^2\right )}{7 d (e (c+d x))^{9/2}} \]
Time = 0.30 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {1117, 1117, 1113, 762}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{\sqrt {-c^2-2 c d x-d^2 x^2+1} (c e+d e x)^{9/2}} \, dx\) |
\(\Big \downarrow \) 1117 |
\(\displaystyle \frac {5 \int \frac {1}{(c e+d x e)^{5/2} \sqrt {-c^2-2 d x c-d^2 x^2+1}}dx}{7 e^2}-\frac {2 \sqrt {-c^2-2 c d x-d^2 x^2+1}}{7 d e (c e+d e x)^{7/2}}\) |
\(\Big \downarrow \) 1117 |
\(\displaystyle \frac {5 \left (\frac {\int \frac {1}{\sqrt {c e+d x e} \sqrt {-c^2-2 d x c-d^2 x^2+1}}dx}{3 e^2}-\frac {2 \sqrt {-c^2-2 c d x-d^2 x^2+1}}{3 d e (c e+d e x)^{3/2}}\right )}{7 e^2}-\frac {2 \sqrt {-c^2-2 c d x-d^2 x^2+1}}{7 d e (c e+d e x)^{7/2}}\) |
\(\Big \downarrow \) 1113 |
\(\displaystyle \frac {5 \left (\frac {2 \int \frac {1}{\sqrt {1-\frac {(c e+d x e)^2}{e^2}}}d\sqrt {c e+d x e}}{3 d e^3}-\frac {2 \sqrt {-c^2-2 c d x-d^2 x^2+1}}{3 d e (c e+d e x)^{3/2}}\right )}{7 e^2}-\frac {2 \sqrt {-c^2-2 c d x-d^2 x^2+1}}{7 d e (c e+d e x)^{7/2}}\) |
\(\Big \downarrow \) 762 |
\(\displaystyle \frac {5 \left (\frac {2 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c e+d x e}}{\sqrt {e}}\right ),-1\right )}{3 d e^{5/2}}-\frac {2 \sqrt {-c^2-2 c d x-d^2 x^2+1}}{3 d e (c e+d e x)^{3/2}}\right )}{7 e^2}-\frac {2 \sqrt {-c^2-2 c d x-d^2 x^2+1}}{7 d e (c e+d e x)^{7/2}}\) |
(-2*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2])/(7*d*e*(c*e + d*e*x)^(7/2)) + (5*(( -2*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2])/(3*d*e*(c*e + d*e*x)^(3/2)) + (2*Ell ipticF[ArcSin[Sqrt[c*e + d*e*x]/Sqrt[e]], -1])/(3*d*e^(5/2))))/(7*e^2)
3.15.5.3.1 Defintions of rubi rules used
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) )*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]
Int[1/(Sqrt[(d_) + (e_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_ Symbol] :> Simp[(4/e)*Sqrt[-c/(b^2 - 4*a*c)] Subst[Int[1/Sqrt[Simp[1 - b^ 2*(x^4/(d^2*(b^2 - 4*a*c))), x]], x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0] && LtQ[c/(b^2 - 4*a*c), 0]
Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_S ymbol] :> Simp[-2*b*d*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(d^2*(m + 1)*(b^2 - 4*a*c))), x] + Simp[b^2*((m + 2*p + 3)/(d^2*(m + 1)*(b^2 - 4*a* c))) Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && LtQ[m, -1] & & (IntegerQ[2*p] || (IntegerQ[m] && RationalQ[p]) || IntegerQ[(m + 2*p + 3) /2])
Leaf count of result is larger than twice the leaf count of optimal. \(338\) vs. \(2(106)=212\).
Time = 4.35 (sec) , antiderivative size = 339, normalized size of antiderivative = 2.69
method | result | size |
default | \(\frac {\left (5 \sqrt {-2 d x -2 c +2}\, \sqrt {d x +c}\, \sqrt {2 d x +2 c +2}\, F\left (\frac {\sqrt {-2 d x -2 c +2}}{2}, \sqrt {2}\right ) d^{3} x^{3}+15 \sqrt {-2 d x -2 c +2}\, \sqrt {d x +c}\, \sqrt {2 d x +2 c +2}\, F\left (\frac {\sqrt {-2 d x -2 c +2}}{2}, \sqrt {2}\right ) c \,d^{2} x^{2}+15 \sqrt {-2 d x -2 c +2}\, \sqrt {d x +c}\, \sqrt {2 d x +2 c +2}\, F\left (\frac {\sqrt {-2 d x -2 c +2}}{2}, \sqrt {2}\right ) c^{2} d x -10 d^{4} x^{4}+5 \sqrt {-2 d x -2 c +2}\, \sqrt {d x +c}\, \sqrt {2 d x +2 c +2}\, F\left (\frac {\sqrt {-2 d x -2 c +2}}{2}, \sqrt {2}\right ) c^{3}-40 c \,d^{3} x^{3}-60 c^{2} d^{2} x^{2}-40 c^{3} d x -10 c^{4}+4 d^{2} x^{2}+8 c d x +4 c^{2}+6\right ) \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}\, \sqrt {e \left (d x +c \right )}}{21 e^{5} \left (d x +c \right )^{4} \left (d^{2} x^{2}+2 c d x +c^{2}-1\right ) d}\) | \(339\) |
elliptic | \(\frac {\sqrt {-e \left (d x +c \right ) \left (d^{2} x^{2}+2 c d x +c^{2}-1\right )}\, \left (-\frac {2 \sqrt {-d^{3} e \,x^{3}-3 c \,d^{2} e \,x^{2}-3 c^{2} d e x -e \,c^{3}+d e x +c e}}{7 d^{5} e^{5} \left (x +\frac {c}{d}\right )^{4}}-\frac {10 \sqrt {-d^{3} e \,x^{3}-3 c \,d^{2} e \,x^{2}-3 c^{2} d e x -e \,c^{3}+d e x +c e}}{21 e^{5} d^{3} \left (x +\frac {c}{d}\right )^{2}}+\frac {10 \left (-\frac {c +1}{d}+\frac {c -1}{d}\right ) \sqrt {\frac {x +\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c -1}{d}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {c -1}{d}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c +1}{d}}}\, F\left (\sqrt {\frac {x +\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c -1}{d}}}, \sqrt {\frac {-\frac {c -1}{d}+\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c}{d}}}\right )}{21 e^{4} \sqrt {-d^{3} e \,x^{3}-3 c \,d^{2} e \,x^{2}-3 c^{2} d e x -e \,c^{3}+d e x +c e}}\right )}{\sqrt {e \left (d x +c \right )}\, \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}}\) | \(397\) |
1/21*(5*(-2*d*x-2*c+2)^(1/2)*(d*x+c)^(1/2)*(2*d*x+2*c+2)^(1/2)*EllipticF(1 /2*(-2*d*x-2*c+2)^(1/2),2^(1/2))*d^3*x^3+15*(-2*d*x-2*c+2)^(1/2)*(d*x+c)^( 1/2)*(2*d*x+2*c+2)^(1/2)*EllipticF(1/2*(-2*d*x-2*c+2)^(1/2),2^(1/2))*c*d^2 *x^2+15*(-2*d*x-2*c+2)^(1/2)*(d*x+c)^(1/2)*(2*d*x+2*c+2)^(1/2)*EllipticF(1 /2*(-2*d*x-2*c+2)^(1/2),2^(1/2))*c^2*d*x-10*d^4*x^4+5*(-2*d*x-2*c+2)^(1/2) *(d*x+c)^(1/2)*(2*d*x+2*c+2)^(1/2)*EllipticF(1/2*(-2*d*x-2*c+2)^(1/2),2^(1 /2))*c^3-40*c*d^3*x^3-60*c^2*d^2*x^2-40*c^3*d*x-10*c^4+4*d^2*x^2+8*c*d*x+4 *c^2+6)/e^5*(-d^2*x^2-2*c*d*x-c^2+1)^(1/2)*(e*(d*x+c))^(1/2)/(d*x+c)^4/(d^ 2*x^2+2*c*d*x+c^2-1)/d
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.49 \[ \int \frac {1}{(c e+d e x)^{9/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=-\frac {2 \, {\left (5 \, {\left (d^{4} x^{4} + 4 \, c d^{3} x^{3} + 6 \, c^{2} d^{2} x^{2} + 4 \, c^{3} d x + c^{4}\right )} \sqrt {-d^{3} e} {\rm weierstrassPInverse}\left (\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right ) + {\left (5 \, d^{4} x^{2} + 10 \, c d^{3} x + {\left (5 \, c^{2} + 3\right )} d^{2}\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} \sqrt {d e x + c e}\right )}}{21 \, {\left (d^{7} e^{5} x^{4} + 4 \, c d^{6} e^{5} x^{3} + 6 \, c^{2} d^{5} e^{5} x^{2} + 4 \, c^{3} d^{4} e^{5} x + c^{4} d^{3} e^{5}\right )}} \]
-2/21*(5*(d^4*x^4 + 4*c*d^3*x^3 + 6*c^2*d^2*x^2 + 4*c^3*d*x + c^4)*sqrt(-d ^3*e)*weierstrassPInverse(4/d^2, 0, (d*x + c)/d) + (5*d^4*x^2 + 10*c*d^3*x + (5*c^2 + 3)*d^2)*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*sqrt(d*e*x + c*e))/ (d^7*e^5*x^4 + 4*c*d^6*e^5*x^3 + 6*c^2*d^5*e^5*x^2 + 4*c^3*d^4*e^5*x + c^4 *d^3*e^5)
\[ \int \frac {1}{(c e+d e x)^{9/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=\int \frac {1}{\left (e \left (c + d x\right )\right )^{\frac {9}{2}} \sqrt {- \left (c + d x - 1\right ) \left (c + d x + 1\right )}}\, dx \]
\[ \int \frac {1}{(c e+d e x)^{9/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=\int { \frac {1}{\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} {\left (d e x + c e\right )}^{\frac {9}{2}}} \,d x } \]
\[ \int \frac {1}{(c e+d e x)^{9/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=\int { \frac {1}{\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} {\left (d e x + c e\right )}^{\frac {9}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{(c e+d e x)^{9/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=\int \frac {1}{{\left (c\,e+d\,e\,x\right )}^{9/2}\,\sqrt {-c^2-2\,c\,d\,x-d^2\,x^2+1}} \,d x \]